I have 4 under the counter cabinet doors that need to be reattached. The old hinges have literally fallen apart. My first problem is finding the right type of hinge. The cabinet doors are made of solid wood, and are partially inset (3/8"). I think a hinge like this one would work.

Here's my bigger problem. The counter is made of brick. the openings for the so-called cabinets are wrapped in 1/8" hard metal. How can I attach these hinges to the metal? I assume I'll need to drill holes. What type of drill bit would I need, and at what diameter? Do I need to thread these holes? Do I need to use really short screws, or should I drill straight into the brick? Or would something like liquid nails alone be able to hold the weight? Any thoughts you may have would be greatly appreciated.

Welcome to the site.
Looking at your picture, it looks like they attached thru the steel into the brick, which should have worked, but the brick broke out.
Glue will not work!
If you can find hinges that fit those holes, you might be able to tap threads in the steel for a machine thread screw, but that might be to big for the holes in the hinges.
If this only a few doors it might be better to line this with wood and buy new doors.
Other may have better ideas.

Another approach might be to drill through the metal with an oversized hole that will be hidden by the hinge. Then use a masonry bit to drill into the brick (but not at a mortar joint), insert an anchor ( such as a Mungo brand plastic anchor, or there are many other types, some steel some plastic ), then screw your hinge screws into the anchor. You need to match the length of the screw to the anchor. So the screw length needs to include the length of the anchor + the metal thickness + the thickness of the hinge plate.

OK, so you are thinking the brick is not solid enough to attach to. Then a variation on your threaded hole idea that might be a little beefier would be to attach a piece of flat bar by means threaded screws or even sheet metal screws, and then attach the cabinet hinges to the flat bar by means of threaded holes.

The old hinges were not attached to the brick in any way. The broken brick in the picture is just a coincidence. There were small holes drilled and threaded, the hinges were held on by very short screws. So I drilled 4 mm holes and tapped them. I used 4 mm machine screws to attach the new hinges. 1 down, 3 to go. Thanks for your advice.